| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Integration with given constant |
| Difficulty | Moderate -0.3 This is a slightly below-average A-level question. Part (a) tests conceptual understanding of the constant of integration (accessible insight), part (b) involves routine integration with chain rule reversal and solving a simple equation, and part (c) requires recognizing a standard substitution (u = sin 2x + 2) but is straightforward once identified. The multi-part structure and 'show detailed reasoning' requirement add some challenge, but all techniques are standard A-level fare with no novel problem-solving required. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| (No because) they differ only by a constant, or eg \(c_2=c_1+\frac{1}{3}\), or \(\frac{1}{3}\) is part of Ben's \(c\). If definite integral found, answers are same. If differentiate, answers same. | B1 | oe, eg They may have different constants of integration. Only the "\(c\)"s are different. Not "Both are correct" or "just different correct methods" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\left[\frac{(1+x)^{-1}}{-1}\right]_1^a\) or \(\left[-\frac{1}{u}\right]_1^{a+1}\) or \(\left[-\frac{1}{\sqrt{u}}\right]_4^{(a+1)^2}\) oe | M1 | Attempt integral, must be of form \(k(1+x)^{-1}\) or \(ku^{-1}\) or \(ku^{-0.5}\) (if from substitution \(u=(1+x)^2\)). Ignore limits |
| \(=\frac{(1+a)^{-1}}{-1}+\frac{1}{2}\) oe | M1 | Attempt substitute appropriate limits into their integral |
| \(\left(=\frac{1}{2}-\frac{1}{1+a}\right) = \frac{a-1}{2(a+1)}\) | A1 | cao oe single fraction |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{a-1}{2(a+1)}=\frac{1}{3}\) or their (b)(i) (limits subst'd) \(=\frac{1}{3}\) | M1 | or their new attempt at \(\int_1^a \frac{1}{(1+x)^2}\,dx=\frac{1}{3}\) |
| \(a=5\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{1}{2}\left[\ln | \sin 2x+2 | \right]_0^{\frac{1}{2}\pi}\) |
| \(=\frac{1}{2}\left[\ln\!\left(\sin\tfrac{1}{6}\pi+2\right)-\ln(0+2)\right]\) | M1 | Attempt substitute both correct limits into their log integral. Allow numerical errors |
| \(=\frac{1}{2}\!\left(\ln\!\left(\tfrac{5}{2}\right)-\ln 2\right)\) | A1 | Allow \(\times\) any \(k\), otherwise any correct form without trig |
| \(=\frac{1}{2}\ln\frac{5}{4}\) oe, eg \(\ln\frac{\sqrt{5}}{2}\) | A1 | Correct one-term exact result. ISW, eg ignore decimal. NB No working, no marks |
# Question 3:
## Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| (No because) they differ only by a constant, or eg $c_2=c_1+\frac{1}{3}$, or $\frac{1}{3}$ is part of Ben's $c$. If definite integral found, answers are same. If differentiate, answers same. | B1 | oe, eg They may have different constants of integration. Only the "$c$"s are different. Not "Both are correct" or "just different correct methods" |
## Part (b)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\left[\frac{(1+x)^{-1}}{-1}\right]_1^a$ or $\left[-\frac{1}{u}\right]_1^{a+1}$ or $\left[-\frac{1}{\sqrt{u}}\right]_4^{(a+1)^2}$ oe | M1 | Attempt integral, must be of form $k(1+x)^{-1}$ or $ku^{-1}$ or $ku^{-0.5}$ (if from substitution $u=(1+x)^2$). Ignore limits |
| $=\frac{(1+a)^{-1}}{-1}+\frac{1}{2}$ oe | M1 | Attempt substitute appropriate limits into their integral |
| $\left(=\frac{1}{2}-\frac{1}{1+a}\right) = \frac{a-1}{2(a+1)}$ | A1 | cao oe single fraction |
## Part (b)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{a-1}{2(a+1)}=\frac{1}{3}$ or their (b)(i) (limits subst'd) $=\frac{1}{3}$ | M1 | or their new attempt at $\int_1^a \frac{1}{(1+x)^2}\,dx=\frac{1}{3}$ |
| $a=5$ | A1 | cao |
## Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{1}{2}\left[\ln|\sin 2x+2|\right]_0^{\frac{1}{2}\pi}$ | M1 | Allow $k\ln(\sin 2x+2)$, $k$ any constant. Ignore limits |
| $=\frac{1}{2}\left[\ln\!\left(\sin\tfrac{1}{6}\pi+2\right)-\ln(0+2)\right]$ | M1 | Attempt substitute both correct limits into their log integral. Allow numerical errors |
| $=\frac{1}{2}\!\left(\ln\!\left(\tfrac{5}{2}\right)-\ln 2\right)$ | A1 | Allow $\times$ any $k$, otherwise any correct form without trig |
| $=\frac{1}{2}\ln\frac{5}{4}$ oe, eg $\ln\frac{\sqrt{5}}{2}$ | A1 | Correct one-term exact result. ISW, eg ignore decimal. NB No working, no marks |
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\begin{enumerate}[label=(\alph*)]
\item Amaya and Ben integrated $( 1 + x ) ^ { 2 }$, with respect to $x$, using different methods, as follows.
Amaya: $\quad \int ( 1 + x ) ^ { 2 } \mathrm {~d} x = \frac { ( 1 + x ) ^ { 3 } } { 3 } + c \quad = \frac { 1 } { 3 } + x + x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } + c$
Ben: $\quad \int ( 1 + x ) ^ { 2 } \mathrm {~d} x = \int \left( 1 + 2 x + x ^ { 2 } \right) \mathrm { d } x = x + x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } + c$
Charlie said that, because these answers are different, at least one of them must be wrong.
Explain whether you agree with Charlie's statement.
\item You are given that $a$ is a constant greater than 1 .
\begin{enumerate}[label=(\roman*)]
\item Find $\int _ { 1 } ^ { a } \frac { 1 } { ( 1 + x ) ^ { 2 } } \mathrm {~d} x$, giving your answer as a single fraction in terms of the constant $a$.
\item You are given that the area enclosed by the curve $y = \frac { 1 } { ( 1 + x ) ^ { 2 } }$, the $x$-axis and the lines $x = 1$ and $x = a$ is equal to $\frac { 1 } { 3 }$.
Determine the value of $a$.
\end{enumerate}\item In this question you must show detailed reasoning.
Find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } \frac { \cos 2 x } { \sin 2 x + 2 } \mathrm {~d} x$, giving your answer in its simplest form.
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2022 Q3 [10]}}